1 Introduction
A large body of research documents the importance of cognitive skills in producing social
and economic success.1 An emerging body of research establishes the parallel importance of
noncognitive skills, i.e., personality, social and emotional traits.2 Understanding the factors
affecting the evolution of cognitive and noncognitive skills is important for understanding
how to promote successful lives.3
This paper estimates the technology governing the formation of cognitive and noncogni-
tive skills in childhood. We establish identification of general nonlinear factor models that
enable us to determine the technology of skill formation. Our multistage technology cap-
tures different developmental phases in the life cycle of a child. We identify and estimate
substitution parameters that determine the importance of early parental investment for sub-
sequent lifetime achievement, and the costliness of later remediation if early investment is
not undertaken.
Cunha and Heckman (2007) present a theoretical framework that organizes and inter-
prets a large body of empirical evidence on child and animal development.4 Cunha and
Heckman (2008) estimate a linear dynamic factor model that exploits cross equation restric-
tions (covariance restrictions) to secure identification of a multistage technology for child
investment.5 With enough measurements relative to the number of latent skills and types of
investment, it is possible to identify the latent state space dynamics generating the evolution
of skills.
The linear technology used by Cunha and Heckman (2008) imposes the assumption that
early and late investments are perfect substitutes over the feasible set of inputs. This paper
identifies a more general nonlinear technology by extending linear state space and factor
analysis to a nonlinear setting. This extension allows us to identify crucial elasticity of sub-
stitution parameters governing the trade-off between early and late investments in producing
adult skills.
Drawing on the analyses of Schennach (2004a) and Hu and Schennach (2008), we es-
1See Herrnstein and Murray (1994), Murnane, Willett, and Levy (1995), and Cawley, Heckman, and
Vytlacil (2001).
2See Heckman, Stixrud, and Urzua (2006), Borghans, Duckworth, Heckman, and ter Weel (2008) and
the references they cite. See also the special issue of the Journal of Human Resources 43 (4), Fall 2008 on
noncognitive skills.
3See Cunha, Heckman, Lochner, and Masterov (2006) and Cunha and Heckman (2007, 2009).
4This evidence is summarized in Knudsen, Heckman, Cameron, and Shonkoff (2006) and Heckman (2008).
5See Shumway and Stoffer (1982) and Watson and Engle (1983) for early discussions of such models.
Amemiya and Yalcin (2001) survey the literature on nonlinear factor analysis in statistics. Our identification
analysis is new. For a recent treatment of dynamic factor and related state space models see Durbin, Harvey,
Koopman, and Shephard (2004) and the voluminous literature they cite.